A Mathematical Model for Breath Gas Analysis of Volatile Organic Compounds with Special Emphasis on Acetone

Noname manuscript No. (will be inserted by the editor)

A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Julian King · Karl Unterkoﬂer · Gerald Teschl · Susanne Teschl · Helin Koc · Hartmann Hinterhuber · Anton Amann

Received: date / Accepted: date

J. King Breath Research Institute of the Austrian Academy of Sciences Dammstr. 22, A-6850 Dornbirn, Austria E-mail: [email protected] K. Unterkoﬂer Vorarlberg University of Applied Sciences and Breath Research Unit of the Austrian Academy of Sciences Hochschulstr. 1, A-6850 Dornbirn, Austria E-mail: karl.unterkoﬂ[email protected] G. Teschl University of Vienna, Faculty of Mathematics Nordbergstr. 15, A-1090 Wien, Austria E-mail: [email protected] S. Teschl University of Applied Sciences Technikum Wien Hochst¨ adtplatz¨ 5, A-1200 Wien, Austria E-mail: [email protected] H. Koc Vorarlberg University of Applied Sciences Hochschulstr. 1, A-6850 Dornbirn, Austria H. Hinterhuber Innsbruck Medical University, Department of Psychiatry Anichstr. 35, A-6020 Innsbruck, Austria A. Amann (corresponding author) Innsbruck Medical University, Univ.-Clinic for Anesthesia and Breath Research Institute of the Austrian Academy of Sciences Anichstr. 35, A-6020 Innsbruck, Austria E-mail: [email protected], [email protected] 2

Abstract Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide (NO) have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These rec- ommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: – breathing patterns affecting gas exchange in the conducting airways – the concentrations in the tracheo-bronchial lining fluid – the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real- time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases and thus is expected to have general relevance for other classes of volatile organic compounds as well. Our model is a first step towards new guidelines for breath gas analysis of acetone (and other water-soluble compounds) similar to those for nitric oxide. Keywords breath gas analysis · volatile organic compounds · acetone · modeling Mathematics Subject Classification (2000) 92C45 · 92C35 · 93C10 · 93B07 3

1 Introduction

Measurement of blood-borne volatile organic compounds (VOCs) occurring in exhaled breath as a result of normal metabolic activity or pathological disorders has emerged as a promising novel methodology for non-invasive medical diagnosis and therapeutic monitoring of disease, drug testing and tracking of physiological processes [2,3,1,61,47]. Apart from the obvious improvement in patient compliance and tolerability, major advantages of exhaled breath analysis compared to conven- tional test procedures, e.g., based on blood or urine probes, include de facto unlim- ited availability as well as rapid on-the-spot evaluation or even real-time analysis. Additionally, it has been pointed out that the pulmonary circulation receives the entire cardiac output and therefore the breath concentrations of such compounds might provide a more faithful estimate of pooled systemic concentrations than single small- volume blood samples, which will always be affected by local hemodynamics and blood-tissue interactions [54]. Despite this huge potential, the use of exhaled breath analysis within a clinical setting is still rather limited. This is mainly due to the fact that drawing reproducible breath samples remains an intricate task that has not fully been standardized yet. Moreover, inherent error sources introduced by the complex mechanisms driving pulmonary gas exchange are still poorly understood. The lack of standardization among the different sampling protocols proposed in the literature has led to the development of various sophisticated sampling systems, which selectively extract end-tidal air by discarding anatomical dead space volume [34,25,13]). Even though such setups present some progress, they are far from being perfect. In particular, these sampling systems can usually not account for the variability stemming from varying physiological states.

In common practice it is tacitly assumed that end-tidal air will reflect the alveolar concentration CA which in turn is proportional to the concentration of the VOC in mixed venous blood Cv¯ , with the associated factor depending on the substance- specific blood:gas partition coefficient λb:air (describing the diffusion equilibrium between capillaries and alveoli), alveolar ventilation V˙A (governing the transport of the compound through the respiratory tree) and cardiac output Q˙c (controlling the rate at which the VOC is delivered to the lungs):

Cv¯ Cmeasured = CA = . (1) V˙A λb:air + Q˙c This is the familiar equation introduced by Farhi [19], describing steady state inert gas elimination from the lung viewed as a single alveolar compartment with a fixed overall ventilation-perfusion ratio V˙A/Q˙c close to one. Since the pioneering work of Farhi, both equalities in the above relation have been challenged. Firstly, for low blood soluble inert gases, characterized by λb:air ≤ 10, alveolar concentrations resulting from an actually constant Cv¯ can easily be seen to vary drastically in response to fluctuations in blood or respiratory flow. While this sensitivity has been exploited in MIGET (Multiple Inert Gas Elimination Technique, cf. [79,78]) to assess ventilation-perfusion inhomogeneity throughout the normal and diseased lung, it is 4 clearly problematic in standard breath sampling routines based on free breathing, as slightly changing measurement conditions (regarding, e.g., body posture, breathing patterns or stress) can have a large impact on the observed breath concentration [17]. This constitutes a typical example for an inherent error source as stated above, poten- tially leading to a high degree of intra- and consequently inter-individual variability among the measurement results [53]. It is hence important to investigate the influence of medical parameters like cardiac output, pulse, breathing rate and breathing volume on VOC concentrations in exhaled breath. In contrast, while highly soluble VOCs (λb:air > 10) tend to be less affected by changes in ventilation and perfusion, measurement artifacts associated with this class of compounds result from the fact that – due to their often hydrophilic properties – a substantial interaction between the exhalate and the mucosa layers lining the conducting airways can be anticipated [6]. In other words, for these substances Cmeasured 6= CA, with the exact quantitative relationship being unknown. Examples of endogenous compounds that are released into the gas phase not only through the blood-alveolar interface, but also through the bronchial lining fluid are, e.g., acetone and ethanol [7,76].

The emphasis of this paper lies on developing a mechanistic specification of end-tidal breath dynamics of highly soluble, blood-borne trace gases during different physiological situations (e.g., rest, exercise, sleep and exposure scenarios) using a prototypic compound. Our main focus will be on the ketone body acetone (2 - propanone; CAS number 67 - 64 - 1; molar mass 58.08 g/mol), which is one of the most abundant VOCs found in human breath and has received wide attention in the literature. Being a natural metabolic intermediate of lipolysis [31], endogenous acetone has been considered as a biomarker for monitoring the ketotic state of diabetic and fasting individuals [73,57,60,66], estimating glucose levels [20] or assessing fat loss [38]. Nominal levels in breath and blood have been established in [80,65], and bioaccumulation has been studied in the framework of exposure studies and pharmacokinetic modeling [84,37,49]. Despite this huge body of experimental evidence, the crucial link between breath and blood levels is still obscure. For instance, multiply- ing the proposed population mean of approximately 1 µg/l [65] in end-tidal breath by the partition coefficient λb:air = 340 [7] at body temperature appears to grossly underestimate observed (arterial) blood levels spreading around 1 mg/l [80,84,33]. Furthermore, continuous profiles of breath acetone extracted during moderate workload ergometer challenges [34] drastically depart from the behavior expected from Equation (1).

In contrast to previous studies which were mainly concerned with the dynamics of single breath tests [76,5,7], our model is simple in the sense that no detailed anatomical features of the respiratory tract are taken into account. Instead, by adopt- ing the usual compartmental approach [37,49], our formulation proves to adequately explain a wide range of experimentally observed data while simultaneously maintaining a model structure amenable to sound mathematical analysis and further algorith- mic treatment in the framework of systems theory. Although such models have been criticized for their underlying simpliﬁcations [21] (e.g., regarding the cyclic nature of breathing), we believe that, if applied in proper way, they provide a practicable bridge 5 between formulations capturing short-term behavior as indicated above and phenomena that are characteristic for sampling scenarios extending over minutes to hours. As previously mentioned the latter typically comprise measurements of VOCs during distinct physiological states such as exercise and are expected to give additional information on the mid- to long-term kinetics of such compounds within various parts of the human body. The physical model to be presented here is driven by the well founded theory covering soluble gas exchange in the single exhalation framework and aims at extending these ideas to a macroscopic level, thereby yielding a model capable of monitoring acetone-speciﬁc distribution processes over longer time periods via the respective breath concentrations.

2 Experimental basics

Breath acetone concentrations are assessed by means of a real-time setup designed for synchronized measurements of exhaled breath VOCs as well as a variety of respiratory and hemodynamic parameters. Our instrumentation has successfully been applied for gathering continuous data streams of these quantities during moderate workload ergometer challenges as well as in a sleep laboratory setting. Details are described elsewhere [34]. In brief, the breath-related part of the mentioned setup consists of a head mask spirometer system allowing for the standardized extraction of predefined exhalation segments which – via a heated and gas tight Teflon transfer line – are then directly drawn into a Proton-Transfer-Reaction mass spectrometer (PTR-MS, Ionicon Analytik GmbH, Innsbruck, Austria) for online analysis. This analytical technique has proven to be a sensitive method for quantification of volatile molecular species X down to the ppb (parts per billion) range on the basis of “soft” chemical ionization within a drift chamber, i.e., by taking advantage of the proton transfer + + H3O + X → XH + H2O from primary hydronium precursor ions originating in an adjoint hollow cathode [41, 42]. Note that this reaction scheme is selective to VOCs with proton affinities higher than water (166.5 kcal/mol), thereby precluding the ionization of the bulk composi- + tion exhaled air, N2,O2 and CO2. Count rates of the resulting product ions XH or fragments thereof appearing at specified mass-to-charge ratios m/z can subsequently be converted to absolute concentrations of the protonated compounds (see [64] for further details on the quantification of acetone as well as [34] for the underlying

PTR-MS settings used). The carbon dioxide concentration CCO2 of the gas sample is determined by a separate sensor (AirSense Model 400, Digital Control Systems, Portland, USA). In addition to the breath concentration profiles of acetone, it will be of importance for us to have at hand a continuous estimate of the corresponding sample water vapor + content Cwater. As has been put forward in the literature, the water dimer (H3O )H2O forming in the drift chamber as a result of clustering can be used for this purpose [82, 4]. More specifically, the pseudo concentration signal at m/z = 37 calculated according to Equation (1) in [64] using a standard reaction rate constant of 2.0×10−9 cm3/s 6 yields a quantity roughly proportional to sample humidity. Slight variations due to fluctuations of the (unknown) amount of water clusters forming in the ion source are assumed to be negligible. Absolute quantification can be achieved by comparison with standards containing predefined humidity levels. Such standards with CCO2 and Cwater varying over the experimental physiological range of 2 - 8 % and 2 - 6 %, respectively, were prepared using a commercial gas mixing unit (Gaslab, Breitfuss Messtechnik GmbH, Harpstedt, Germany), resulting in a mean calibration factor of 2.1×10−4 and R2 ≥ 0.98 for all regressions. It has been argued in [32] that the aforementioned pseudo concentration can drastically be affected by the carbon dioxide concentration, which, however, could not be confirmed with our PTR-MS settings.

Although the computed water content is slightly overestimated with increasing CCO2 , the sensitivity was found to stay within 10 % of the mean value given above. Nev- ertheless, we recognize that this approximate method for determining water vapor levels can only serve as a ﬁrst surrogate for more exact hygrometer measurements. Despite the fact that molecular oxygen is not protonated, the breath oxygen concentration CO2 – relative to an assumed steady state value of about 100 mmHg in end-tidal (alveolar) air during rest [44] – within one single experiment can still be + assessed by monitoring the parasitic precursor ion O2 at m/z = 32 (adjusted to be + below 2 % of H3O ), resulting from a small amount of sample gas entering the ion source with subsequent ionization of O2 under electron impact [54]. For normalization purposes, the respective count rates are again converted to pseudo concentrations. Table 1 summarizes the measured quantities relevant for this paper. In general, breath concentrations will always refer to end-tidal levels, except where explicitly noted. Moreover, an underlying sampling interval of 5 s is assumed for each variable.

Table 1 Summary of measured parameters together with some nominal literature values during rest and assuming ambient conditions. Breath concentrations refer to end-tidal levels.

Variable Symbol Nominal value (units)

Cardiac output Q˙c 6 (l/min) [48]

Alveolar ventilation V˙A 5.2 (l/min) [83] Tidal volume VT 0.5 (l) [83]

Acetone concentration Cmeasured 1 (µg/l) [65]

CO2 content CCO2 5.6 (%) [44] Water content Cwater 4.7 (%) [23]

O2 content CO2 13.7 (%) [44]

3 Acetone modeling

3.1 Preliminaries and assumptions

Classical pulmonary inert gas elimination theory [19] postulates that uptake and removal of VOCs take place exclusively in the alveolar region. While this is a reasonable assumption for low soluble substances, it has been shown by several authors that exhalation kinetics of VOCs with high afﬁnity for blood and water such as acetone 7

are heavily inﬂuenced by relatively quick absorption and release mechanisms occurring in the conductive airways, see, e.g., Ref. [5]. More speciﬁcally, due to their pronounced hydrophilic characteristics such compounds tend to interact with the water- like mucus membrane lining this part of the respiratory tree, thereby leading to pre- and post-alveolar gas exchange often referred to as wash-in wash-out behavior. The present model aims at taking into consideration two major aspects in this framework.

3.1.1 Bronchial exchange

It is now an accepted fact, that the bronchial tree plays an important role in overall pulmonary gas exchange of highly (water) soluble trace gases, affecting both endogenous clearance as well as exogenous uptake. For instance, Anderson et al. [7] inferred that while fresh air is being inhaled, it becomes enriched with acetone stored in the airway surface walls of the peripheral bronchial tract, thus leading to a decrease of the acetone pressure/tension gradient between gas phase and capillary blood in the alveolar space. This causes an effective reduction of the driving force for gas exchange in the alveoli. In contrast, during exhalation the aforementioned diffusion process is reversed, with a certain amount of acetone being stripped from the airstream and rede- posited onto the previously depleted mucus layer, thereby decreasing overall acetone elimination as compared to purely alveolar extraction. Similarly, exposition studies suggest a pre-alveolar absorption of exogenous acetone during inhalation and a post- alveolar revaporization during expiration, resulting in a diminished uptake of highly soluble VOCs than what would be expected if exchange occurred completely in the alveoli [84,37,74].

In accordance with previous modeling approaches, we consider a bronchial compartment separated into a gas phase and a mucus membrane, which is assumed to inherit the physical properties of water [37,49] and acts as a reservoir for acetone. Part of the acetone dissolved in this layer is transferred to the bronchial circulation, whereby the major fraction of the associated venous drainage is postulated to join the pulmonary veins via the postcapillary anastomoses [44], cf. Fig. 2. A study by Mor- ris et al. [51] on airway perfusion during moderate exercise in humans indicates that the fraction qbro ∈ [0,1) of cardiac output Q˙c contributing to this part of bronchial perfusion will slightly decrease with increasing pulmonary blood ﬂow. According to ˙rest Figure 3 in that contribution and assuming that Qc = 6 l/min we can derive the heuristic linear model

˙ rest ˙ ˙rest qbro(Qc) := max{0,qbro (1 − 0.06(Qc − Qc ))}. (2)

rest The constant qbro will be estimated in Section 4. As a rough upper bound we propose rest the initial guess qbro = 0.01 [44]. We stress the fact that the bronchial compartment just introduced has to be interpreted as an abstract control volume lumping together the decisive sites of airway gas exchange in one roughly homogenous functional unit. These locations can be expected to vary widely with the solubility of the VOC under scrutiny as well as with physiological boundary conditions [5]. 8

3.1.2 Temperature dependence

The solubility of acetone in the mucus surface of the airways is strongly affected by temperature, which in turn can be assessed by taking into account absolute sample humidity as follows. Passing through the conditioning regions of the upper airways, inhaled air will be warmed to a mean body core temperature of approximately 37 ◦C and fully saturated with water vapor, thus leading to an absolute humidity of bronchial and alveolar air of about 6.2 % at ambient pressure. During exhalation, depending on the axial temperature gradient between the lower respiratory tract and the airway opening, a certain amount of water vapor will condense out and reduce the water content Cwater in the exhalate according to the saturation water vapor pressure Pwater (in mbar) determined by local airway temperature T (in ◦C) [46,24]. The relationship between these two quantities can be approximated by the well-known Magnus formula [68] 17.62T P (T) = 6.112 exp , (3) water 243.12 + T valid for a temperature range −45 ◦C ≤ T ≤ 60 ◦C. For normal physiological values of T, the resulting pressure is sufﬁciently small to treat water vapor as an ideal gas [59] and hence by applying Dalton’s law we conclude that absolute humidity Cwater (in %) of the exhalate varies according to

Pwater(T) Cwater(T) = 100 , (4) Pambient

where Pambient is the ambient pressure in mbar. Inverting the above formula, the mini- mum airway temperature Tmin = Tmin(Cwater) during exhalation becomes a function of measured water content in exhaled breath. From this, a mean airway and mucus temperature characterizing the homogenous bronchial compartment of the previous section will be deﬁned as T (C ) + 37 T¯(C ) := min water , (5) water 2 corresponding to a hypothesized linear increase of temperature along the airways. Note that this assumption is somehow arbitrary in the sense that the characteristic temperature of the airways should be matched to the primary (time- and solubility- dependent) location of airway gas exchange as mentioned above. Equation (5) thus should only be seen as a simple ad hoc compromise incorporating this variability.

The decrease of acetone solubility in the mucosa – expressed as the water:air partition coefficient λmuc:air – with increasing temperature can be described in the ambient temperature range by a van’t Hoff-type equation [70] B log λ (T) = −A + , (6) 10 muc:air T + 273.15 where A = 3.742 and B = 1965 Kelvin are proportional to the entropy and enthalpy of volatilization, respectively. Hence, in a hypothetical situation where the absolute sample humidity at the mouth is 4.7 % (corresponding to a temperature of T ≈ 32 ◦C 9 and ambient pressure at sea level, cf. [46,23]), local solubility of acetone in the mucus ◦ layer increases from λmuc:air(37 C) = 392 in the lower respiratory tract (cf. [36]) to ◦ λmuc:air(32 C) = 498 at the mouth, thereby predicting a drastic reduction of airstream acetone concentrations along the airways. The above formulations allow to assess this reduction by taking into account sample water vapor as a meta parameter. This meta parameter reflects various influential factors on the mucus solubility λmuc:air which would otherwise be intricate to handle due to a lack of information, such as local airway perfusion, breathing patterns, mucosal hydration and thermoregulatory events which in turn will affect axial temperature profiles. In particular, λmuc:air for the entire bronchial compartment will be estimated via the mean airway temperature T¯ as

λmuc:air(Cwater) = λmuc:air(T¯(Cwater)). (7) The strong coupling between sample humidity and exhaled breath concentrations predicted by the two relationships (3) and (6) is expected to be a common factor for all highly water soluble VOCs. In the framework of breath alcohol measurements Lindberg et al. [40] indeed showed a positive correlation between these two quantities along the course of exhalation, which can also be observed in the case of acetone, cf. Fig. 1. Variations of the acetone blood:air partition coefficient λb:air = 340 [7,18] – dom- inating alveolar gas exchange – in response to changes in mixed venous blood temperature, e.g., due to exercise, are ignored as such changes are necessarily small [16]. ◦ Hence, λb:air will always refer to 37 C. Similarly, the partition coefficient between mucosa and blood is treated as constant defined by

◦ λmuc:b := λmuc:air(37 C)/λb:air, (8) resulting in a value of 1.15. Note, that if the airway temperature is below 37 ◦C we always have that λmuc:air/λmuc:b ≥ λb:air, (9) as λmuc:air is monotonically decreasing with increasing temperature, see Equation (6).

3.1.3 Bronchio-alveolar interactions

In a series of modeling studies [76,5], the location of gas exchange (from gas phase to liquid phase and vice versa) has been demonstrated to shift between bronchial and alveolar regions depending on the solubility of the compound under investigation. While during tidal breathing exchange for substances with blood:air partition coefﬁcient λb:air ≤ 10 takes place almost exclusively in the alveoli, it appears to be strictly limited to the bronchial tract in the case of λb:air ≥ 100. Transport for VOCs lying within these two extremes distributes between both spaces. Likewise, for ﬁxed λb:air, location of gas exchange is expected to vary with breathing patterns. As has been concluded by Anderson et al. [7], airway contribution to overall pulmonary exchange of endogenous acetone is about 96 % during tidal breathing, but only 73 % after inspiration to total lung capacity. The rationale for this reduction is that while more proximal parts of the mucosa lining are being depleted earlier in the course of inhalation by loosing acetone to the inhalate, saturation of the airstream with acetone 10

800 10

8 600

6 400 [%] [ppb] 4

200 Acetone [ppb] 2 Water [%] 0 0 0 5 10 15 20

400

1500 exhalation volume [ml] exhalation flow rate [ml/s] 300

1000 200 [ml] [ml/s]

500 100

0 0 0 5 10 15 20 [s]

800

700

600

500

Acetone [ppb] r2 = 0.9431 400

300 4.5 5 5.5 6 6.5 7 Water [%]

Fig. 1 Correlation between breath acetone concentrations and breath water content during free exhalation at an approximately constant rate of 150 ml/s. Water-acetone pairs in the third panel correspond to the two linear Phase 3 segments as described in [7]. The high sampling frequency is achieved by limiting the PTR-MS measurement cycle to only three mass-to-charge ratios (with corresponding dwell times given in brackets): m/z = 21 (5 ms), m/z = 37 (2 ms) and m/z = 59 (10 ms). is continuously shifted towards the alveolar region. Furthermore, it can be argued that the magnitude of this shift increases with volumetric flow during inhalation as equilibration of fresh air with the mucus layer in regions with high flow rates might not be completed. From a reversed viewpoint this would be consistent with the observation made in [7] that end-exhaled acetone partial pressures increase with exhaled flow rate.

Such smooth transitions in the location of gas exchange can be incorporated into the model by including a diffusion process describing the interaction between bronchial and alveolar compartment (cf. Fig. 2), which is similar to the strategy found in the theory of stratiﬁed or series inhomogeneities developed by Scheid et al. [62]. According to the approach presented there, mentioned inhomogeneities arise 11

from the fact that while gas flow in the upper parts of the respiratory tree is pri- marily dominated by convection, axial diffusion becomes the decisive factor in the terminal airspaces. This can also be thought of as incomplete mixing of tidal volume with the functional residual capacity, thus leading to an effective increase of alveolar deadspace. It will be shown that reinterpreting stratified inhomogeneity and the stratified conductance parameter D (having units of volume divided by time and taking values in [0,∞)) in the framework of soluble gas exchange allows a proper description of bronchio-alveolar interactions. The alveolar region itself is represented by one single homogenous alveolar unit, thereby neglecting ventilation-perfusion inequality throughout the lung. In the case of VOCs with high λb:air this constitutes an acceptable simplification, since the classical Farhi equation predicts a minimal influence of local ventilation-perfusion ratios on the corresponding alveolar concentrations. Uptake and elimination of VOCs to and from the bronchio-alveolar tract during inhalation and exhalation is governed by the alveolar ventilation V˙A.

3.1.4 Body compartments

The systemic part of the model has been adapted from previous models [36,49] and consists of two functional units: a liver compartment, where acetone is endogenously produced and metabolized, as well as a tissue compartment representing an effective storage volume. The latter basically lumps together tissue groups with similar blood:tissue partition coefficient λb:tis ≈ 1.38, such as richly perfused tissue, mus- cles and skin [7,49]. Due to the low fractional perfusion of adipose tissue and its low affinity for acetone, an extra fat compartment was not considered. The fractional blood flow qliv ∈ [0,1) to the liver is assumed to be related to total cardiac output by ˙ ˙ ˙rest qliv(Qc) := 0.034 + 1.145exp(−1.387Qc/Qc ), (10) obtained by exponential fitting of the data given in [49]. The rate M˙ of acetone metabolism is assumed to be proportional to liver concentration, i.e., 0.75 M˙ = klinbw Clivλb:liv =: kmetClivλb:liv, (11) where bw denotes the body weight in kg and the kinetic rate constant klin is obtained by linearization of the Michaelis-Menten kinetics proposed in [36], i.e., we expect a value of 0.75 klin := vmax/km ≈ 0.0037 l/kg /min. (12) Taking into account nominal mixed venous blood concentrations of approximately 1 mg/l for healthy volunteers as well as an apparent Michaelis-Menten constant km = 84 mg/l this will result in a good approximation for most situations encountered in practice (except for extreme cases, e.g., severe diabetic ketoacidosis or starvation ketosis where plasma concentrations up to 500 mg/l have been reported [57,60] and hence metabolism can be expected to reach saturation). Other ways of acetone clearance such as excretion via the renal system are neglected [84]. Endogenous synthesis of acetone in the liver is assumed to occur at some rate kpr > 0 depending on current lipolysis [31]. In particular, fat catabolism is considered a long-term mechanism compared to the other dynamics of the system, so that kpr can in fact be assumed constant during the course of experiments presented here (less than 2 hours). 12

3.2 Model equations and a priori analysis

3.2.1 Derivation

In order to capture the gas exchange and tissue distribution mechanisms presented above, the model consists of four different compartments. A sketch of the model structure is given in Fig. 2 and will be detailed in following.

CI Cbro 6 V˙A ? Cbro Cmuc bronchial q compartment Vbro Vmuc bro Q ˙ c 6D - ? ( CA Cc0 1

− alveolar

q compartment VA Vc0 bro ) Q ˙

6 c

Cliv,b ˙ qliv(1 − qbro)Qc

Vliv,b ( 1 liver −

- q compartment Cliv kmet liv )(

Vliv 1 kpr − q bro ) Q ˙ C c tis tissue Vtis compartment

Fig. 2 Sketch of the model structure. The body is divided into four distinct functional units: bronchial/mucosal compartment (gas exchange), alveolar/end-capillary compartment (gas exchange), liver (metabolism and production) and tissue (storage). Dashed boundaries indicate a diffusion equilibrium.

Model equations are derived by taking into account standard conservation of mass laws for the individual compartments, see Appendix A.1.2. Local diffusion equilib- ria are assumed to hold at the air-tissue, tissue-blood and air-blood interfaces, the ratio of the corresponding concentrations being described by the appropriate partition coefﬁcients, e.g., λb:air. Unlike for low blood soluble compounds, the amount of highly soluble gas dissolved in local blood volume of perfused compartments cannot generally be neglected, as it might signiﬁcantly increase the corresponding capaci- ties. This is particularly true for the airspace compartments. Since reliable data for some local blood volumes could not be found, in order not to overload the model 13

with too many hypothetical parameters, we will use the effective compartment volumes V˜bro := Vbro +Vmucλmuc:air, V˜A := VA +Vc0 λb:air, V˜liv := Vliv +Vliv,bλb:liv as well as V˜tis := Vtis and neglect blood volumes for the mucosal and tissue compartment, see also Appendix A.1.2. According to Fig. 2 as well as by taking into account the discussion of the previous subsections, for the bronchial compartment we ﬁnd that

dCbro λmuc:air V˜bro = V˙A(CI −Cbro) + D(CA −Cbro) + qbroQ˙c Ca − Cbro , (13) dt λmuc:b

with CI denoting the inhaled (ambient) gas concentration, while the mass balance equations for the alveolar, liver and tissue compartment read dC A V˜ = D(C −C ) + (1 − q )Q˙ C − λ C , (14) dt A bro A bro c v¯ b:air A and dC liv V˜ = k − k λ C + q (1 − q )Q˙ C − λ C , (15) dt liv pr met b:liv liv liv bro c a b:liv liv and dC tis V˜ = (1 − q )(1 − q )Q˙ C − λ C , (16) dt tis liv bro c a b:tis tis respectively. Here,

Cv¯ := qlivλb:livCliv + (1 − qliv)λb:tisCtis (17) and Ca := (1 − qbro)λb:airCA + qbroλmuc:airCbro/λmuc:b (18) are the associated concentrations in mixed venous and arterial blood, respectively. Moreover, we state that measured (end-tidal) breath concentrations equal bronchial levels, i.e., Cmeasured = Cbro. (19)

The decoupled case D = qbro = 0 will be excluded further on in this paper as it lacks physiological relevance.

3.2.2 Steady state relationships and interpretation of the stratiﬁed conductance parameter D

Assume that the system is in steady state and we know the associated end-tidal breath concentration Cmeasured = Cbro. Furthermore, we deﬁne the bronchial and alveolar ventilation-perfusion ratio to be

V˙A V˙A rbro := and rA := , qbroQ˙c (1 − qbro)Q˙c respectively. If D = 0 we deduce that

Cmeasured = Cbro = r C + (1 − q )λ C r C + (1 − q )C r C +C bro I bro b:air A = bro I bro v¯ = bro I a , (20) λmuc:air λmuc:air λmuc:air (1 − qbro) + rbro (1 − qbro) + rbro + rbro λmuc:b λmuc:b λmuc:b 14

corresponding to purely bronchial gas exchange. On the other hand, for D → ∞ it can be shown by simple algebra that

Cmeasured = Cbro = rACI +Cv¯ Ca CA = = . (21) λmuc:air λmuc:air qbro + (1 − qbro)λb:air + rA qbro + (1 − qbro)λb:air λmuc:b λmuc:b

In the following, let CI = 0. Note that then in both cases the physiological boundary condition Cv¯ ≥ Ca is respected. Substituting qbro = 0 into (21) yields the well-known Farhi equations describing purely alveolar gas exchange. Thus, D deﬁnes the location of gas exchange in accordance with Section 3.1.3, while qbro determines to what extent the bronchial compartment acts as an inert tube. In this sense, Equations (13)–(16) deﬁne a generalized description of gas exchange including several known models as special cases. This hierarchy is summarized in Fig. 3.

model

qbro = 0 @ qbro > 0 © @@R stratiﬁed location dependent inhomogeneity gas exchange

D → ∞ D → 0 D → ∞ ? ? ? Farhi inert gas bronchial alveolar elimination

Fig. 3 Equations (13)–(16) viewed as generalized model including several gas exchange mechanisms as special cases (CI = 0).

For perspective – as has been rationalized in the case of acetone in Section 3.1.3 and is likely to be a common characteristic for all highly water soluble VOCs – the stratified conductance parameter D will be close to zero during rest and is expected to increase with tidal volume and/or flow rate [77,5]. With hindsight, as will be justified in Section 4.2, we propose to model this dependency as rest ˙ ˙ rest D := kdiff,1 max{0,VT −VT } + kdiff,2 max{0,VA −VA }, kdiff,i ≥ 0. (22)

We stress the fact that if D is close to zero, calculating blood levels from Cmeasured by using Equation (20) or taking advantage of this expression for normalization and correction purposes is an ill-posed problem due to the possibly large inﬂuence of the term rbro as well as the uncertainty regarding qbro and λmuc:air. Particularly, from the aforementioned facts this means that measured breath concentrations of acetone (and generally highly water soluble substances) determined during resting conditions and free breathing can be rather misleading indicators for endogenous levels, even if 15

sampling occurs under well defined standard conditions, for instance – as is common practice – using CO2- and/or flow-controlled extraction from the end-tidal exhalation segment [34,13]. This has first been recognized in the context of breath alcohol measurements revealing experimentally obtained blood-breath concentration ratios of ethanol during tidal breathing that are unexpectedly high compared with in vitro partition coefficients [30]. An elegant approach proposed to circumvent this problem is isothermal rebreathing [30,55], which aims at creating conditions where (cf. Equation (20))

rbroCbro + (1 − qbro)Cv¯ Cv¯ Ca Cmeasured = CI = CA = Cbro = = = (23) (1 − qbro)λb:air + rbro λb:air λb:air from which the mixed venous blood concentration can easily be determined by multi- plying the measured rebreathing concentration with the anticipated blood:air partition coefﬁcient at body temperature. Further discussion of this technique is postponed to another paper (in preparation). Another idea to arrive at a more reliable estimation of endogenous acetone levels obviously might be to mimic steady states corresponding to a situation where D is large, so that Equation (21) can be assumed to hold approximately. The natural experimental framework here are workload challenges as described in Section 4.2. However, a general word of caution is in order at this point, certainly limiting the practical value of aforementioned steady state calculations. Firstly, experimental data in breath gas analysis usually corresponds to a transient state of the system, despite the fact that the breath concentration itself might appear to be reasonably constant, compare, e.g., with Fig. 4. Moreover, one equilibrium alone does usually not provide enough information to uniquely determine all endogenous variables of interest. Thus, estimation has to be based on dynamic behavior rather than steady states, which will be the topic of Section 4.

3.2.3 A priori analysis

Equations (13)–(16) can be written as a time-varying linear inhomogeneous ODE system c˙ = A(u,θ)c + b(u,θ) =: g(u,c,θ) (24) T in the state variable c := (Cbro,CA,Cliv,Ctis) , which is dependent on a time-independent parameter vector θ as well as on a vector u := (V˙A,Q˙c,VT,λmuc:air(Cwater),CI) lumping together all measured variables. The associated measurement equation reads

y = Cmeasured = (1,0,0,0)c =: h(c). (25) In the present paragraph we are going to discuss some qualitative properties of the system presented. Firstly, as the system is linear, for any given initial condition c(0) there is a unique global solution. Moreover, A is a Metzler matrix, meaning that all off-diagonal entries are non-negative and b ≥ 0. Consequently ci = 0 implies that c˙i ≥ 0 for every component and it follows that the trajectories remain non-negative. Thus, (24) constitutes a positive system with the state c evolving within the positive n n orthant R>0 := {c|c ∈ R , ci > 0, i = 1,...,n}, where n = 4. 16

n Proposition 1 All solutions of (24) starting in R>0 remain bounded.

Proof This can be shown by considering the total mass m := ∑V˜ici ≥ 0 and noting that m˙ = kpr − kmetλb:livCliv +V˙A(CI −Cbro). (26) Taking into account positivity of the solutions and the involved parameters this shows that Cliv is bounded from above for bounded CI, since the assumption that Cliv is unbounded yields a contradiction. Analogously, Cbro and CA can be shown to be bounded by consideringm ˜ := V˜broCbro +V˜ACA +V˜tisCtis and similarly for Ctis.

Furthermore, it can be proven that under physiological steady state conditions, i.e., for constant u, the above system has a globally asymptotically stable equilib- −1 rium c0 := −A b. To this end it sufﬁces to show that the time-invariant matrix A is Hurwitz, i.e., all real parts of the associated eigenvalues are negative.

Proposition 2 Suppose u is time-independent. Then the real parts of all eigenvalues of A(u,θ) are non-positive. They are strictly negative if det(A(u,θ)) 6= 0. Moreover, det(A(u,θ)) = 0 if and only if either Q˙c vanishes or two of the quantities V˙A,qliv,kmet vanish.

Proof For this it is sufficient to confirm that A is diagonally dominant, i.e., there exists a vector z > 0 such that the row vector zT A is non-negative. This is a simple consequence of the fact that in such a case A can be shown to be similar to a matrix A˜ which has the claimed property. Indeed, if we define the diagonal matrix U := −1 diag(z1,...,zn) and set A˜ := UAU , it holds that

ai jzi 1 T a˜ j j + ∑ |a˜i j| = a j j + ∑ = (z A) j ≤ 0 (27) i6= j i6= j z j z j for all j since A is Metzler. Moreover, note thata ˜ j j < 0 for all j. Hence, the ﬁrst claim will follow from Gershgorin’s circle theorem. The required vector z follows from

(V˜bro,V˜A,V˜liv,V˜tis)A = (−V˙A,0,−kmetλb:liv,0).

Moreover, the only case when the real part of an eigenvalue can vanish is when it lies on the boundary of a circle touching the imaginary axis, that is, when an eigenvalue is zero. This proves the ﬁrst part. To show the second part one veriﬁes that det(A) = Q˙c (ϑ1V˙A + ϑ2kmet)qlivQ˙c + ϑ3V˙Akmet (28) with ϑ j > 0.

This completes the minimal set of properties which necessarily should to be sat- isﬁed in any valid model of concentration dynamics. In particular, global asymptot- ical stability for the autonomous system ensures that – starting from arbitrary initial values – the compartmental concentrations will approach a unique equilibrium state once the physiological inputs u affecting the system are ﬁxed. Obviously, this is of 17 paramount importance when aiming at the description of processes exhibiting pronounced steady states (which in the context of breath gas analysis corresponds, e.g., to the situation encountered during rest or constant workload [34]) and a prerequisite for orchestrating reproducible experiments. Hence, in the context of VOC modeling, any approach not incorporating this property will lack a fundamental characteristic of the observed data.

Remark 1 We stress the fact that if the rate of metabolism is described by saturation rather than linear kinetics, i.e., if the term kmetClivλb:liv in Equation (15) is replaced by v C λ M˙ = max liv b:liv , (29) km +Clivλb:liv with vmax, km > 0, the system becomes essentially nonlinear. However, all conclu- sions drawn so far remain generally valid. Firstly, by applying the Mean Value The- orem note that the Michaelis-Menten term above is Lipschitz and so again a global Lipschitz property holds for the right-hand side of the resulting ODE system. Pos- itivity and boundedness of the solutions for arbitrary inputs u can be established in analogy with the arguments in the last paragraph. Effectively, in order to show boundedness from above, we ﬁrst note that Equation (26) now reads

vmaxClivλb:liv m˙ = kpr − +V˙A(CI −Cbro) (30) km +Clivλb:liv and as a result Cbro is unconditionally bounded if V˙A > 0. From Equation (13) it then follows that CA must be bounded and subsequently Cv¯ (and hence Cliv as well as Ctis) are bounded using Equation (14). In the case V˙A = 0, the inequality

kpr ≤ vmax (31) is necessary and sufﬁcient for boundedness. Necessity is easily deduced from the fact that kpr > vmax impliesm ˙ > 0 and consequently m (i.e., at least one component ci) is unbounded. Conversely, Equation (31) ensures that Cliv is bounded and hence the same arguments as in the linear case apply. In order to establish the existence of a globally asymptotically stable equilibrium point for ﬁxed u we adopt a result on monotone systems taken from Leenheer et. al. [39], see also [29]. For monotone systems in general with applications to biological systems we refer to [67,8]. Firstly, note that the steady state relation associated with Equation (15) now becomes

2 0 = Cliv + a1Cliv + a0 (32) with a0 < 0. Thus, there exists a unique positive steady state solution for the liver concentration Cliv and consequently the same can be conﬁrmed to hold true for the other components of c as well. As a consequence, it follows that the model has a n unique equilibrium point in the non-negative orthant M := R≥0. Moreover, note that ∂g i ≥ 0 for i 6= j, i, j ∈ {1,...,n}, (33) ∂c j 18

i.e., the system is cooperative. This term stems from the fact that a given component is positively affected by the remaining ones. For linear systems, an equivalent characterization is that the corresponding system matrix A is Metzler. A well-known n result on this type of systems asserts that the associated semiﬂow Φ : R≥0 × R → n R , (t,c0) 7→ Φt (c0) := c(t) is monotone with respect to the natural (componentwise) n partial order on R given by

c ≤ z if and only if ci ≤ zi, i ∈ {1,...,n}. That is, Φ preserves the order of the initial conditions (see [67], Prop. 3.1.1), i.e., n for c0,c˜0 ∈ R the condition c0 ≤ c˜0 implies that Φt (c0) ≤ Φt (c˜0) for t ∈ [0,∞). As a third requirement, since all trajectories are bounded and the system evolves within n the closed state space M ⊂ R it follows that for every c ∈ M the corresponding semi- orbit O(c) := {Φt (c), t ≥ 0} has compact closure. In summary, we are now in the situation to apply Theorem 5 of [39] (by choosing X = M), which asserts that the aforementioned properties are sufﬁcient for the unique equilibrium point in M to be globally attractive.

One of the main purposes of the proposed model is to provide a basis for estimating (unknown) compartment concentrations c as well as certain acetone-specific rest parameters θi ∈ {kpr,kmet,vmax,km,kdiff,1,kdiff,2,qbro } from the knowledge of measured breath concentrations y. Often over-looked, a necessary requirement in this framework is a priori (or structural) identifiability/observability of the model, which basically checks that in an ideal context of an error-free (i.e., “correct”) model and noise-free measurements there exist functions u (or, in other words, conductible experiments) such that the associated output y (the accessible data) carries enough information to – at least locally – enable an unambiguous determination of all unknown states and parameters. Particularly, this avoids an over-parameterization of the model. As the time evolution of the system (13)–(16) for a given u is fixed once the initial conditions c0 at the start of the experiment are known, the analysis of a priori identifiability/observability hence amounts to studying (local) injectivity of y with respect to c0 and the parameters θi under scrutiny. Evidently, if such a property does not hold then any attempt to reliably estimate these quantities is doomed to failure from the start, as two entirely different parameter combinations can yield exactly the same data. In the present context, generic a priori identifiability/observability for c0 and any selection of parameters θ j as above was confirmed by interpreting the latter as additional states with time derivative zero and subjecting the augmented (possibly nonlinear, but control-affine) system to the Hermann-Krener rank criterion [26] (see Appendix A.2), which can be seen as a nonlinear analog of the Kalman observability criterion. Intuitively, generic a priori identifiability/observability guarantees that “almost every” experiment represented by some choice of u allows to distinguish the “true” values of c0 and θ j from any other possible choice lying close to these values by giving rise to a unique process response. It is evident that generic local observability is indeed the best we can hope for in our context. For instance, looking at the metabolizing liver compartment of the model (13)–(16), if we set θ1 := kpr and θ2 := kmet and interpret them as additional states having time derivative zero, the augmented model obviously will not be locally observable if c0 is an equilibrium state 19

associated with some constant input u. This is simply due to the fact that kpr and kmet cannot be determined simultaneously from the single steady state equation related to Equation (15) and constitutes a typical example where the matrix J in (49) becomes rank-deﬁcient for a speciﬁc choice of the initial conditions.

4 Model validation and estimation

4.1 Simulation of exposure data and model calibration

Validation and refinement of the decisive but uncertain nominal values for the kinetic rate constants of metabolism and production, the fractional bronchial blood flow as well as for the equilibrium tissue levels c0 at rest is carried out by fitting the corresponding model response to the Series 1 acetone exposure scenario studied by Wigaeus et al. [84,49]. This in vivo benchmark data set consists of pooled observed acetone concentration profiles in exhaled breath, arterial and venous blood of eight healthy male volunteers during and after exposure to CI = 1.3 mg/l acetone over a pe- riod of two hours. We assume that the exposure starts after ten minutes of quiet tidal breathing, i.e., the inhaled concentration is given by the scaled indicator function CI(t) = 1.3 χ[10,130](t) and set alveolar ventilation and cardiac output to constant rest- ˙ rest ˙rest ing values VA = 6 l/min and Qc = 5.8 l/min, respectively [49,36]. Tissue volumes and partition coefficients are as in Table 3 for a male of height 180 cm and weight 70 kg. Furthermore, acetone metabolism is assumed to follow a Michaelis-Menten kinetics with a fixed apparent Michaelis constant km = 84 mg/l (see Section 3.1.4). rest rest Our aim is to determine the parameter vector θ = (vmax,kpr,D ,qbro ) as well as the nominal endogenous steady state levels c0 by solving the ordinary least squares problem  g(u0,c0,θ) = 0 (steady state) n  2  c0,θ ≥ 0 (positivity) argmin C , j −C (t j) , s.t. c0,θ ∑ a a qrest ≤ 0.05 (normalization) j=1  bro Ca(0) = 1 mg/l (endogenous arterial level)

Here, g is the right-hand side of the ODE system (13)–(16), Ca, j is the measured arterial blood concentration at time instant t j, whereas the predicted arterial concentration Ca is deﬁned as in Equation (18). The proposed endogenous level of Ca(0) = 1 mg/l is in accordance with other values given in the literature [31,84]. The above minimization problem was solved by implementing a multiple shooting routine [15] in Matlab. This iterative method can be seen as a generalization of the standard Gauss- Newton (i.e., single shooting) algorithm, designed to avoid divergence issues of the latter due to large residuals. For further details as well as convergence and stability properties we refer to [14,58]. The necessary derivatives of the trajectories with respect to (c0,θ) were computed by simultaneously solving the associated variational equations [22]. Starting values were chosen to be uniformly distributed in the interval [0,1] and convergence was assumed to be achieved when the maximum componentwise relative change between two successive parameter reﬁnements was less than 20

0.01 %. Fig. 4 shows that the calibrated model is capable of convincingly reproduc- ing the pooled data given in [84].

20 1.4

C 18 bro 1.2 C I 16 data

1 14

12 0.8 [mg/l] 10 I [mg/l] , C a C

bro 0.6

8 C

6 0.4

4 C a 0.2 2 data

0 0 0 50 100 150 200 250 0 50 100 150 200 250 [min] [min]

Fig. 4 Model ﬁtted to the Series 1 exposure data published by Wigaeus et al.. First panel: observed versus predicted arterial concentrations. Second panel: observed versus predicted breath concentrations Cmeasured = Cbro.

Table 2 Fitted parameter values for the Series 1 exposure data published by Wigaeus et al..

rest rest vmax kpr D qbro (Cbro,CA,Cliv,Ctis) (mg/min/kg0.75) (mg/min) (l/min) (%) (mg/l) Anticipated 0.31 ≤ 2 0 1 (0.0011,∗,∗,∗) Optimized 0.62 0.19 0 0.43 (0.0016,0.0029,0.58,0.72)

Moreover, literature or anticipated values of c0 and θ compare favorably to those obtained by the aforementioned minimization, cf. Table 2. In particular, the ﬁtted value of the gas exchange location parameter Drest = 0 agrees well with what is expected from the theoretical discussion in Section 3.2.2. The rate at which endogenous acetone forms in healthy adults as a result of normal fat catabolism is not known. However, from the published study on acetone metabolism in lean and obese humans during starvation ketosis by Reichard et al. [60] a rather conservative upper bound for kpr can be derived to lie in the range of 2 mg/min. A diminished fractional bronchial rest perfusion qbro might reﬂect the fact that the assumption of diffusion equilibrium be- 21

tween the mucosa lining and the deeper vascularized sections of the airway wall is rather stringent, so that actually less acetone is transported away from the peripheral rest bronchial tract via the bloodstream. On the other hand qbro = 0.01 refers to the entire bronchial circulation rather than only the part contributing to anatomic right-to-left shunt as discussed in Section 3.1.1. The fitted steady state value for the measured breath concentration Cmeasured(0) = Cbro(0) = 0.0016 mg/l during normal breathing at rest is only slightly higher than the observed levels spreading around 500 ppb (corresponding to about 0.0011 mg/l) [65,34]. The population spread of the fitted parameters within the study cohort could be assessed, e.g., by a Bayesian [50] or mixed effects approach [35], which, however would be beyond the scope of this paper. Here, the major aim rather is to demonstrate the flexibility of the model in covering a wide spectrum of different experimental scenarios.

Fig. 4 illustrates the necessity of taking into account the conducting airways as an additional compartment for exchange of highly soluble substances. In particular, the observed data profiles sharply contrast the classical Farhi inert tube description predicting arterial blood concentrations Ca to be directly proportional to (alveolar) breath concentrations CA = Cmeasured. On the other hand, from the viewpoint of pre- alveolar uptake as discussed in Section 3.1.1, accumulation of exogenous acetone in the systemic circulation is expected to be delayed due to the small contribution of bronchial blood flow to overall perfusion. 0.75 In the following, the fitted value of klin = vmax/km = 0.0074 l/kg /min is interpreted as an intrinsic property of acetone metabolism in the liver, while inter-individual variation is introduced by multiplication with the 0.75 power of body weight. Hence, kmet as in Equation (11) is viewed as a constant known parameter. Similarly, for rest identifiability reasons qbro will be fixed at its fitted value 0.0043, cf. Section 4.2.

4.2 Ergometer datasets and comparison with other models

A primary motivation for this work was to develop a model capable of explaining observed acetone profiles in exhaled breath resulting from endogenous levels (i.e., while breathing of an atmosphere with negligible acetone content) during moderate workload ergometer challenges, cf. [34]. As has been discussed there, end-tidal acetone levels closely resemble the profile of alveolar ventilation respectively inhalation volume, showing abrupt increases respectively drops in the range of 10 – 40 % at the onsets and stops, respectively, of the individual workload periods, see Fig. 5 (a), first panel. Similarly, a series of auxiliary experiments carried out by means of the same instrumental setup showed that increasing tidal volume during rest results in increased breath acetone concentrations while increasing respiratory frequency has a less pronounced impact, cf. Fig. 5 (b). Both effects appear to support the hypothesis of Section 3.1.3 that acetone exchange is strongly influenced by volume and speed of inhalation and hence suggest that any model not incorporating this mechanism will fail to reproduce these results. In particular, from the viewpoint of the classical Farhi description, the profile in Fig. 5 is rather counter-intuitive. For instance, during hyperventilation, acetone supply from the bloodstream tends to stay constant or rise 22

only slightly, while a drastically increased ventilation enhances dilution of the alveolar air and therefore is expected to decrease the corresponding breath concentration. Indeed, it can be shown that the Farhi model, which corresponds to a special case of our model by setting qbro = 0 and D → ∞ is not capable of recovering the measured proﬁles.

(a) Exercise 75 W (b) Hyperventilation

breath acetone concentration present model Farhi et al. / Mörk et al. 1.8

1.6 g/l] µ [ 1.4 1.2

0 5 10 15 20 25 30

3 C [µg/l] C [µg/l] 0.8 bro A Cliv [mg/l] Ctis [mg/l] 2.5

g/l] 0.6 µ [ 2 [mg/l]

1.5 0.4

0 5 10 15 20 25 30

80 4 ^ alveolar ventilation [l/min] D [l/min] tidal volume [l] 60

40 2 [l] [l/min] 20

0 0 0 5 10 15 20 25 30

5 15 cardiac output [l/min] ventilation−perfusion ratio [ ] 4

3 [ ]

[l/min] 10 2

1 5 0 5 10 15 20 25 30

6 550 water content [%] mucus−air partition coefficient [ ] 5 500 [ ] [%] 4 450

3 400 0 5 10 15 20 25 30 [min]

Fig. 5 Typical proﬁle of end-exhaled acetone concentrations in response to the following physiological regime: rest (0-3 min), ergometer challenge at 75 W (3-12 min), rest (12-20 min), hyperventilation with increased tidal volume (20-22 min), rest (22-25 min), high-frequency hyperventilation (25-28 min), rest (28-30 min). 23

The same holds true for the physiological compartment model introduced by Mork¨ et al. [49], which has been designed for the purpose of explaining acetone exhalation kinetics during rest and exercise and has been validated for exposure scenarios at hand of the Wigaeus et al. datasets presented in the last section. The model of Mork¨ et al. is implemented by replacing the mass balance Equations (13)–(14) of the respiratory tract with Equations (4)–(7) of that paper. The body compartments remain unchanged. Additional parameter values are adopted as given in [49]. Initial compartment concentrations as well as kpr for the models of Mork¨ et al. and Farhi et al. model are calculated by solving the associated steady state equations at t0 = 0 given CI ≡ 0 and Cbro(0) = 1.3 µg/l. Similarly, the response for the present model is computed by solving the ordinary least squares problem  n g(u0,c0,kpr,α) = 0 (steady state) 2  argmin y −C (t ) , s.t. c ,k ,α ≥ 0 (positivity) c0,kpr,α ∑ j bro j 0 pr j=0 Cbro(0) = 1.3 µg/l (initial level) on the time interval [t0,tn] = [0,tmax]. Here, y j = Cmeasured, j is the observed acetone concentration at time instant t j and α = (α1,...,αm−1) is a coefﬁcient vector for the piecewise linear spline function

 t−si−1 t ∈ [si−1,si] m−1  si−si−1 ˆ si+1−t D(t) := αiSi(t), Si(t) := t ∈ [si,si+1] ∑ si+1−si i=1  0 otherwise

used for approximating the time-varying stratified conductance parameter D ∈ [0,∞) on m subintervals covering the time span [t0 = s0,tmax = sm]. The nodes si are chosen to result in an equidistant partition of about 0.5 min, cf. Fig. 5, third panel. For simulation purposes the measured physiological functions are converted to input function handles u by applying a local smoothing procedure to the associated data and interpo- lating the resulting profiles with splines. The aforementioned optimization problem was solved as described in Section 4.1. Fig. 5 summarizes the results of these calculations. The visually good fit can formally be assessed by residual analysis. Plots of the resulting residuals versus time and versus model predictions clearly exhibit random patterns, suggesting that the assumptions of i.i.d., homoscedastic additive measurement errors underlying ordinary least squares methodology might be reasonable [11]. Furthermore, no statistically significant autocorrelation among the residuals or cross- correlation between the residuals and the measured inputs could be detected, indicating that the model has picked up the decisive dynamics underlying the data [43].

While all three models will describe the steady state at rest, only Equations (13)– (16) tolerably capture the entire observable dynamical behavior. The second panel in Fig. 5 clearly reveals the physiological mechanism underlying the step-shaped dynamics of breath acetone concentrations in response to constant load exercise and hyperventilation: during rest and tidal breathing, measured (bronchial) levels markedly differ from the alveolar ones due to the effective diffusion barrier between the two spaces represented by a value of Drest ≈ 0. As soon as tidal volume and/or respiratory frequency are increased, this barrier – as reflected in the fitted profile of Dˆ 24

– will vanish to some extent according to the processes discussed in Section 3.2.2, causing Cbro = Cmeasured to approach a value closer to CA. In contrast, CA itself as well as the breath acetone profiles simulated by means of the other two models remain relatively constant. This is consistent with the behavior expected from the Farhi formulation, predicting minimal sensitivity of alveolar concentrations with respect to fluctuations of the ventilation-perfusion ratio for highly soluble trace gases. In other words, the major part of short-term variability observable in breath acetone concentrations during free breathing can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. In particular, note that with the present model the observed acetone dynamics can again be captured by applying a constant endogenous production rate of approximately 0.17 mg/min. The above-mentioned reasoning appears to agree with previous observations in the literature, where the excretion (defined as the ratio between steady state partial pressures in mixed expired air and mixed venous blood) of acetone has been demonstrated to increase during moderate exercise [63].

The high degree of interplay between Dˆ and V˙A as well as VT discernible in the third panel of Fig. 5 suggests to characterize the time-dependency of D via these two respiratory variables, e.g., as proposed in Equation (22). By employing non-negative least square methods we find that kdiff,1 = 17.67 and kdiff,2 = 0.81 with the associated model again giving a good agreement with the observed data. This parameterization might be used to reduce the originally infinite dimensional estimation problem for D to two additional parameters. While in Section 3.2.3 it has been confirmed that the model is structurally locally observable with respect to the initial conditions and rest θ ⊆ {kpr,kmet,kdiff,1,kdiff,2,qbro }, it remains to be investigated if the data corresponding to exercise or hyperventilation tests as presented above is rich enough to admit a joint numerical estimation of all these variables. This amounts to studying the a posteriori identifiability (also termed practical identifiability or estimability) of the model given the specific experimental situation. For an overview of related concepts we refer to [28]. Intuitively, it should be clear from the approximately constant profile of Cliv that kpr and kmet can hardly be assessed simultaneously, as both values are coupled via a single steady state equation associated with Equation (15). Using rest similar reasoning, smaller values of qbro (leading to larger steady state concentrations CA(0)) can be compensated for by smaller constants kdiff,i to yield an almost identi- cal model output. A more formal investigation of this situation can be carried out by calculating the correlation coefficients between the sensitivities of the model output with respect to these parameters. Values of these indices associated with (kmet,kpr) rest rest as well as (kdiff,1,qbro ) and (kdiff,2,qbro ) are close to 1 or -1, thereby revealing a substantial degree of collinearity between the corresponding sensitivities and providing strong indications that a proper estimation of any of these pairs on the basis of moderate exercise challenges cannot be anticipated, cf. Appendix A.2. This observation rest is clearly problematic, as particularly qbro and the two diffusion constants kdiff,i have to be regarded as highly uncertain parameters which are not accessible by direct measurement. In a subsequent paper, we will review isothermal rebreathing as a powerful experimental technique for mitigating this type of ill-conditionedness. 25

4.2.1 Discussion and critical remarks

The main intention of this section is to critically review and clarify some of the assumptions underlying the model derivation as well as to indicate some potential im- provements of the present formulation.

Diffusion equilibrium in the bronchial tract. While it is commonly agreed upon that the transport of inert gases between the alveolar space and end-capillary blood is perfusion-limited [78], a similar premise in the case of airway gas exchange remains less certain. Experimental and theoretical evidence appears to favor the view that although a diffusion equilibrium might be attained at the air-mucus interface [76, 37], the bronchial epithelial tissue can constitute an effective diffusion barrier between the mucus lining and the bronchial circulation. The magnitude of this diffusional resistance is probably substance-specific, with an inverse relation to molecular weight [72]. In this sense, the fractional bronchial blood flow qbro has to be interpreted as effective perfusion of those mucosal tissue layers for which an instanta- neous equilibrium with air can be achieved [37]. Specifically, qbro might differ for distinct compounds. While in the special case of acetone animal models seem to support the assumption of a complete equilibration between airstream and bronchial circulation [72], in order to extend the validity of the model over a wider range of highly water soluble VOCs it might hence be necessary to include a diffusion limita- tion between these two compartments. The statistical significance of this generalization might then be assessed by employing residual based comparison techniques for nested models as described in [11,10]. However, at the current stage of research and given the limited data on the behavior of breath trace gases having similar physico- chemical characteristics like acetone, we prefer to maintain a parameterization as parsimonious as possible.

Continuous ventilation and temperature dependence. Due to the tidal nature of breathing, bronchial as well as alveolar gas concentrations will vary throughout the breathing cycle, following a roughly periodical pattern during normal breathing. These variations can be captured by considering two separate mass balance systems, describing the dynamics of the associated concentrations for each inhalation and exhalation phase, respectively [45,37,5]. While such microscopic formulations are of paramount importance for resolving events within one individual respiratory cycle, when looking at the mid- to long-term behavior of breath VOCs we are rather interested in the global dynamics of the averaged compartmental concentrations. This approach leads to models of continuous ventilation with a unidirectional gas stream and has the enormous applicatory advantage of reducing the model structure to one single mass balance system. Similarly, variable temperature distributions in the bronchial compartment are represented by a single mean temperature T¯ according to Equation (5), thereby lumping together our ignorance regarding the exact temperature proﬁle along the airways. Different functional relationships might be investigated here.

In summary, this paper introduces a novel compartmental description of pulmonary gas exchange and systemic distribution of blood-borne, highly blood and 26 water soluble VOCs, which faithfully captures experimentally determined end-tidal acetone profiles for normal healthy subjects during free breathing in distinct physiological states. Additionally, the model has been validated in the framework of an external exposure scenario published in the literature and illuminates the discrepan- cies between observed and theoretically predicted blood-breath ratios of acetone during resting conditions, i.e., in steady state. In this sense, the present model provides a good compartmental perspective of acetone exhalation kinetics and is expected to contribute to a better understanding of distribution, transport, biotransformation and excretion processes of acetone in different functional units of the organism as well as their impact on the observed breath concentration. The emphasis of this work has been laid on deriving a sound mathematical formulation flexible enough to cover a wide spectrum of possible VOC behavior, while simultaneously maintaining physiological plausibility as well as a clear-cut physiological interpretation of the involved parameters. Care has been taken to keep the parameterization as parsimonious as possible, thereby constructing a well-founded “minimal” model respecting fundamental physiological boundary conditions, such as boundedness of the involved dynamic components and the existence of a globally asymptotically stable equilibrium state. While a complete sensitivity and practical identifiability analysis was beyond the scope of this paper and has to be matched to the particular experimental framework in which the model will be used, general concepts from structural and practical identifiability have been exploited in order to provide some indications regarding the information content of the observable breath level with respect to the endogenous situation.

Acknowledgements The research leading to these results has received funding from the European Com- munity’s Seventh Framework Programme (FP7/2007-13) under grant agreement No. 217967. Julian King is a recipient of a DOC fellowship at the Breath Research Institute of the Austrian Academy of Sciences. Anton Amann greatly appreciates the generous support of the Member of the Tyrolean regional government Dr. Erwin Koler and the Director of the University Clinic of Innsbruck (TILAK) Mag. Stefan Deﬂorian. Gerald Teschl acknowledges support from the Austrian Science Fund (FWF) under Grant No. Y330.

A Appendix

A.1 Physical preliminaries

The following paragraphs brieﬂy summarize the fundamental physical principles underlying the present model derivation. Further discussion largely follows standard textbooks on thermodynamics and biological mass transport; see for instance [75].

A.1.1 Diffusion and Henry’s Law

Fick’s ﬁrst law of diffusion states that the net ﬂux J quantifying diffusional transport of a gas species X between two spatially homogenous control volumes (i.e., compartments in which X can be postulated to 0 behave uniformly) A and A is proportional to the associated difference in partial pressure PX , i.e.,

J = kdiff(PX,A − PX,A0 ), (34) where the non-negative diffusion constant kdiff has dimensions of amount (in mol) divided by pressure and time. If the gas under scrutiny is in dissolved state rather than in gas phase, PX is referred to as gas tension, 27

deﬁned as the partial pressure that would be exerted by the gas above the liquid if both were in equilibrium (e.g., oxygen tension in blood). In the following we will omit the subscript X. Equation (34) can also be expressed via a difference in concentrations. We will distinguish the following three cases: (i). Firstly, if A as well as A0 represent a gaseous medium and X can be treated as an ideal gas (which will be a general premise in the following) then, by the ideal gas law, P can be written as, e.g.,

PA = CART, (35) where CA is the concentration in A and R and T denote the gas constant and absolute temperature, respectively. Hence, ˜ J = kdiff(CART −CA0 RT) = kdiff(CA −CA0 ), (36)

where k˜diff is deﬁned by the above equation. (ii). Similarly, if both A and A0 are liquids, we can make use of Henry’s law stating that the amount of a gas X that can dissolve in a liquid at a particular temperature is directly proportional to the respective tension, viz., C = HP. (37) Here the solute- and solvent-speciﬁc Henry constant H = H(T) is inversely related to temperature. Sub- stituting the last expression into Equation (34) yields

CA CA0 J = kdiff − . (38) HA HA0 (iii). Finally, let us assume that A0 represents a liquid, while A is a gas volume. The prototypic example for this situation in our context is the blood-gas interface between respiratory microvasculature and the alveoli. In this case by combining (35) and (37) we arrive at

CA0 CA0 J = kdiffRT CA − = k˜diff CA − . (39) HA0 RT HA0 RT In case of a diffusion equilibrium, i.e., J = 0, we deduce from Equation (38) that

CA0 HA0 = =: λA0:A, (40) CA HA

0 where the positive dimensionless quantity λA0:A is the so-called partition coefﬁcient between A and A (e.g., λblood:fat). Accordingly, the reciprocal value is denoted by λA:A0 . Analogously, from (39) the ratio between the liquid phase concentration and gas phase concentration of X in equilibrium is given by

CA0 = HA0 RT. (41) CA

The partition coefﬁcient HA0 RT is again denoted by λA0:A (e.g., λblood:air) and is a measure of the solubility of the gas X in the solvent A0, thereby allowing a classiﬁcation of X as low or highly soluble with respect to A0.

A.1.2 Compartmental mass transport

0 As above, consider two homogenous compartments A and A with volumes VA and VA0 , respectively. Gas dynamics within these two compartments are governed by conservation of mass, stating that the rate at which the compartmental amount (e.g., VACA) of X changes per time unit equals the rate of transfer into the compartment less the rate of removal from the compartment. For instance, the process of diffusion between A and A0 as discussed before can be seen as a special case of this scheme with J (expressed in terms of concentrations) combining both diffusional inflow and outflow. Hence, if A and A0 interact by diffusion and both are affected by additional input rates I˙ and effluent output rates O˙ (which might also comprise possible sources or sinks, respectively), then, assuming that the respective compartment volumes remain constant over time, the mass balances read dC V A = −J + I˙ − O˙ (42) A dt A A 28

and dCA0 V 0 = J + I˙ 0 − O˙ 0 , (43) A dt A A respectively. If the diffusion constant is sufficiently large to make J the clearly dominant term in the above equations, i.e., if diffusion is fast compared to the other dynamics influencing the system, CA as well as CA0 will instantaneously approach their steady state values and a permanent diffusion equilibrium can 0 be assumed to hold between A and A . In this case, by replacing CA0 in (43) with CAλA0:A according to Equation (40), we see that the ODE system above simplifies to the one-dimensional differential equation

dCA V 0 λ 0 +V = I˙ 0 − O˙ 0 + I˙ − O˙ . (44) A A :A A dt A A A A

The factor (VA0 λA0:A +VA) can be seen as “effective” volume V˜A of the combined compartment. In the framework of physiological modeling this practice is routinely followed when setting up mass balance equations describing the kinetics of an inert gas X within a functional entity of the human body composed of two or more subcompartments. For perspective, consider the alveolar tract where the branching capillary network of the respiratory microvasculature promotes a fast equilibration between end-capillary blood A0 and gas phase A even under moderate exercise conditions [78]. In other words, the transport of X within the alveolar tract is perfusion-limited rather than diffusion-limited. A common error here is to generally neglect the contribution of VA0 λA0:A to the effective alveolar volume as VA0 ≈ 0.15 l is argued to be much smaller than VA ≈ 3 l. Note, however, that this is only acceptable for low soluble inert gases with blood:air partition coefﬁcient less than 1. Similarly, for most tissue groups, owing to their high degree of vascu- larization the intracellular space A can be assumed to rapidly equilibrate with the extracellular space A0 (including the vascular blood and the interstitial space). In this case, a venous equilibrium is said to hold, i.e., the concentration CA0 of X in blood leaving the tissue group is related to the actual tissue concentration CA via Equation (40).

Note that the assumption of well-mixing is central to compartmentalization, and can often only be justiﬁed heuristically, either by examining factors that contribute to rapid distribution (e.g., convection) or by considering small volumes. If heterogeneity within a compartment is expected to be substantial or the main focus is on the spatial distribution of X, general mass transport equations leading to PDEs have to be employed.

A.2 A primer on local observability

For the convenience of the reader, the following section serves as a self-contained overview of the concept of (local) observability for analytic, single-output state-space models which are afﬁne with respect to a given set u = (u1,...,up) of bounded input functions u j : [0,T] → R, j = 1,..., p, i.e., systems of the form p x˙ = g0(x) + ∑ g j(x)u j, x(0) = x0, j=1 (45) y = h(x).

n Here it is assumed that x(t) evolves within an open set M ⊆ R , h : M → R is real analytic, and the n g j : M → R are analytic vector ﬁelds. We specialize to the case where the inputs are constant and hence can be interpreted as additional known parameters of the system

x˙ = g(x), x(0) = x0, (Σ) y = h(x).

Furthermore, it is required that a unique solution exists, e.g., by assuming that g is globally Lipschitz on M. Since g is assumed to be analytic, the solution will also be analytic with respect to both t and x0. The output y = y(t,x0) is viewed as a function of the initial condition x0. It will be analytic as well. Equations (Σ) are a valid description for many biological processes under constant measurement conditions and represent a sufﬁcient framework for the type of models and experiments considered here. Further exposition follows the treatment in [26,52,69] and [9]. 29

Intuitively, observability (or a priori identiﬁability of the initial condition seen as unknown parameter of the model) characterizes the possibility to unambiguously determine x0 (and hence the entire dynamics of x) from the exact knowledge of y, i.e., in the ideal context of error-free measurements. In particular, local observability states that x0 can be instantaneously distinguished from any of its neighbors by using the system response. This is made explicit in the following deﬁnition.

Deﬁnition 1 Let V ⊆ M. Two initial conditions x0,x˜0 ∈ V are V-indistinguishable (x0 ∼V x˜0) if for every T > 0 such that the corresponding trajectories remain in V it holds that

y(t,x0) = y(t,x˜0) for all t ∈ [0,T]. The initial conditions x0 and x˜0 are called indistinguishable if they are V-indistinguishable for any such V. A system is called locally observable at x0 ∈ M, if there exists a neighbourhood V ⊆ M of x0 such that all points in V are V-indistinguishable.

In other words, local observability characterizes the fact that x0 7→ y(.,x0) is injective for arbitrarily small T. For the sake of illustration, consider the system

    x˙1 −(x2 + x3)x1 x˙2 =  0 , y = x1, (46) x˙3 0

then y = x1,0 exp(−(x2,0 + x3,0)t).

Evidently, this toy model is not locally observable as the slightly perturbed initial conditionsx ˜2,0 := x2,0 +ε andx ˜3,0 := x3,0 −ε result in exactly the same response y for any ε > 0. If we interpret x1 and x2 as constant parameters, the above situation corresponds to a typical case of overparameterization, i.e., adding x3 does not provide a more detailed description of the observable output. On the other hand, from the reversed viewpoint of estimation, this means that we will never be able to simultaneously estimate both x2,0 and x3,0 from the available process data but can only assess their sum. An equivalent characterization of this fact can be given in the context of sensitivities, i.e., the partial derivatives of the output with respect to the parameters. Indeed, we immediately see that

∂y ∂y = , ∂x2,0 ∂x3,0 i.e., the sensitivities are linearly dependent, which is well-known to preclude a joint estimation of both parameters. Intuitively, the reason for this is that from the the Taylor expansion of ﬁrst order

∂y ∂y y(x˜2,0,x˜3,0) ≈ y(x2,0,x3,0) + ε − ε = y(x2,0,x3,0), ∂x2,0 ∂x3,0

which again conﬁrms that any small change in x2,0 can be compensated for by an appropriate change in x3,0 to yield almost the same output. A rigorous statement of this fact in the general unconstrained case is given in [12]. In contrast, as will be illustrated henceforth, local observability guarantees linear independence of the derivatives of y with respect to x0, which also is a necessary requirement for the broad class of numerical parameter estimation schemes based on the minimization of a given cost functional (e.g., in ordinary least squares methodology) using ﬁrst-order information.

In more complex models, where an analytical expression for y can usually not be derived, we have to resort to differential-geometric methods for investigating local observability. These tools will only be developed to the extent necessary for stating the final observability criterion, but are usually defined in a much more general framework than the one presented here. For this purpose, first note that using the chain rule, the time derivative of the output equals the Lie derivative Lgh of the output function h with respect to the vector field g, i.e.,

y˙ = (∇h) · g =: Lgh 30

which again is analytic. Here, ∇ = ( ∂ ,..., ∂ ) denotes the gradient and · denotes the scalar product in ∂x1 ∂xn n (0) (k) (k−1) R . Similarly, by setting Lg h = h and iteratively deﬁning Lg h = LgLg h for k ≥ 1 we immediately see that k (k) ∂ (k) y (x0) := y(t,x0) = Lg h(x0). ∂tk t=0 By our analyticity assumption on the system (Σ), the system response y is analytic and hence y can be represented by a convergent power (Lie) series around t = 0:

∞ k ∞ k t (k) t (k) y(t,x0) = ∑ y (x0) = ∑ Lg h(x0). (47) k=0 k! k=0 k!

As a result, x0 will only be V-distinguishable from x˜0 for any V if we can ﬁnd at least one k ≥ 0 with the property that (k) (k) Lg h(x0) 6= Lg h(x˜0). This motivates the following two deﬁnitions. The linear space over R of all iterated Lie derivatives

( ) = {L(k)h( ), k ≥ }, O x0 : spanR g x0 0 is called the observation space of the model. Moreover, let ∇O := {∇H, H ∈ O}. The famous Hermann- Krener criterion [26] now states that the model is locally observable at x0 if

dim∇O(x0) = n. (48)

Note that the sufficient condition (48) is equivalent to finding n indices k j, j = 1,...,n such that the (k j) gradients ∇Lg h(x0) are linearly independent. In fact, it turns out [9] that only the first n iterated Lie derivatives have to be computed in order to decide if such indices exist. Summing up yields a simple algebraic rank criterion for local observability at a fixed point x0, namely

(0) T (n−1) T rank J(x) := rank ∇Lg h(x0) ,...,∇Lg h(x0) = n. (49)

Note that for a linear system

x˙ = Ax, y = Cx

the above condition simpliﬁes to rank CT ,AT CT ,...,A(n−1)T CT = n,

which is the well-known criterion introduced by Kalman (see, e.g., [69]). Obviously, if the linear model is locally observable at one particular initial condition, local observability holds for every other initial condition as well. To conclude this section let us assume that g and h belong to the field of rational functions in x = (x1,...,xn) over R, i.e., g,h ∈ R(x1,...,xn) with no real poles in M. Such rational models constitute an acceptable simplification for most biological processes describing mass balance systems and including standard kinetic mechanisms (e.g., Michaelis-Menten terms). Consequently, from the structure of the model (45) the entries of the matrix J defined in (49) belong to R(x,u) and hence the rank of J(x,u) can be defined as the maximum size of a submatrix whose determinant is a non-zero rational function. This so-called generic rank can easily be determined by symbolic calculation software. Hence, if the generic n+p rank of J is n, the rank over R of J(x,u) evaluated at one specific point (x,u) ∈ R will also be n except for certain singular choices that are algebraically dependent over R. As these special points constitute a set of measure zero, we can conclude that if the generic rank of J(x,u) equals n, almost every experiment corresponding to a choice of constant input variables u will suffice to immediately distinguish a particular unknown initial condition x0 from its neighbors by giving rise to a distinct system output y. Obviously, this is a necessary prerequisite in any attempt to identify x0 from measured data. This powerful result is strengthened further by applying the Universal Input Theorem for analytic systems due to Sussmann [71], 31

ensuring that if a pair of unknown initial conditions can be distinguished by some (e.g., constant) input, it can also be distinguished by almost every input belonging to C ∞([0,T]), T > 0 (see also [81] for a rigorous statement as well as a self-contained proof). Note, that in the context of physiological models this is the appropriate class of functions for capturing a wide range of applicable physiological conditions.

Note that the line of argumentation remains valid for analytic g and h. However, the generic rank in this case usually has to be estimated, e.g., by computing the rank of J evaluated at some randomly n+p assigned points (x,u) ∈ R . While from the above an affirmative outcome of the rank test guarantees that every initial condition within a neighborhood of the “true” value x0 will yield a distinct output y when conducting a generic experiment with smooth inputs, the effective reconstruction of x0 from partially observed and error-corrupted data remains a challenging problem. For instance, despite a positive result of the rank criterion, nothing can be said about the degree of linear independence between the derivatives of the output with respect to the initial conditions. In fact, the associated sensitivity matrix can still be rank-deficient in a numerical sense, indicating a high correlation among some of the components to be recovered. This is the context of practical identifiability or estimability by means of numerical estimation schemes, which will particularly depend on the specific experimental situation. 32

A.3 Nomenclature

Table 3 Basic model parameters and nominal values during rest; LBV denotes the lean body volume in liters calculated according to LBV = −16.24 + 0.22bh + 0.42bw, with body height (bh) and weight (bw) given in cm and kg, respectively [49].

Parameter Symbol Nominal value (units) Compartment concentrations

bronchioles Cbro, Cmeasured 1 (µg/l) [65] alveoli CA arterial Ca 1 (mg/l) [84,31] mixed-venous Cv¯

liver Cliv tissue Ctis inhaled (ambient) CI Compartment volumes

bronchioles Vbro 0.1 (l) [49] mucosa Vmuc 0.005 (l) [49]

alveoli VA 4.1 (l) [49]

end-capillary Vc0 0.15 (l) [27] liver Vliv 0.0285 LBV (l) [49] blood liver Vliv,b 1.1 (l) [56]

tissue Vtis 0.7036 LBV (l) [49] Fractional blood ﬂows

fractional flow bronchioles qbro 0.01 [44] fractional flow liver qliv 0.32 [49] Partition coefficients at body temperature

blood:air λb:air 340 [7,18]

mucosa:air λmuc:air 392 [70,36] blood:liver λb:liv 1.73 [36] blood:tissue λb:tis 1.38 [7] Metabolic and diffusion constants 0.75 linear metabolic rate kmet 0.0074 (l/kg /min) [fitted] 0.75 saturation metabolic rate vmax 0.31 (mg/min/kg ) [36] apparent Michaelis constant km 84 (mg/l) [36] endogenous production kpr 0.19 (mg/l) [fitted] stratified conductance D 0 (l/min) [fitted]

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